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arXiv · 1711.01460

On rational singularities and counting points of schemes over finite rings

Itay Glazer
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Abstract

We study the connection between the singularities of a finite type $\mathbb{Z}$-scheme X and the asymptotic point count of X over various finite rings. In particular, if the generic fiber $X_{\mathbb{Q}}=X\times_{\mathrm{Spec}\mathbb{Z}}\mathrm{Spec}\mathbb{Q}$ is a local complete intersection, we show that the boundedness of $\frac{\left|X(\mathbb{Z}/p^{n}\mathbb{Z})\right|}{p^{n\mathrm{dim}X_{\mathbb{Q}}}}$ in p and n is in fact equivalent to the condition that $X_{\mathbb{Q}}$ is reduced and has rational singularities. This paper completes a result of Aizenbud and Avni.

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Itay Glazer. 2017-11-04. On rational singularities and counting points of schemes over finite rings. https://doi.org/10.2140/ant.2019.13.485

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